Percolation threshold, critical

The percolation simulations clearly allow ring formation it also represents self-avoiding statistics, i.e., includes the excluded volume effects in good solvents. Finally, the probability of placing units on lattice sites becomes more and more dependent on whether a site in the neighborhood is already occupied. In other words the percolation experiment becomes a non-mean field approach when the occupation reaches the critical percolation threshold. Therefore, strong deviations were expected between the more accurate percolation and the Flory-Stockmayer mean field approaches. Physicists were of the opinion that the mean field results must be basically wrong. 

Mixing different cross-linkers (19b and 19c) yielded systems with a strong gel-weak gel transition, rather than a distinct sol-gel shift. When the concentration of each of the cross-linkers was above the critical percolation threshold, the kinetically slower cross-linker (19c) dictated the gel properties. Upon addition of enough of a competitive binding additive, such as DMAP, to drop the concentration of the active cross-linking units below their individual percolation thresholds but still allowing the total amount of both active cross-linkers (19b and 19c) to be above the percolation threshold results in a gel whose properties are now controlled by the kinetically faster cross-linker (19c). 

As seen from Table 10.1 impressive oxygen fluxes have been reported through 25 mol% yttria-stabilized bismuth oxide (BY25) [110] and 25 mol% erbia-stabilized bismuth oxide (BE25) [111,112], which oxide electrolytes were rendered electronically conductive by dispersion with silver metal. A prerequisite is that both constituent phases in the composite membranes do form a continuous path for both ionic and electronic conduction, having their concentrations above the critical (percolation threshold) volume fraction <])(,. The latter quantity determines the minimum volume fraction in which conduction is possible and is a function of, for example, the relative dimensions and shape of the particles of both constituent phases [113]. In actual composite materials,... 

In general, at a mass loading of about 50% PANI, the volume fraction of the conducting PANI was well above the critical percolation threshold level of around 16% [113] so that conductivity as high as 1 S/cm should have been expected. The lower conductivity values ( 10 3 S/cm) consistently observed in the PANI-MMT composites possibly resulted from the presence of low conducting MMT in the materials. In fact, entry 5, Table 4 shows a much higher conductivity for the PANI prepared under the same conditions without any MMT. 

Extensive empirical studies of percolation networks have revealed universal critical percolation thresholds (pc). That is, for percolation to occur across a network, some critical proportion of the squares (or blocks in three-dimensions) must be pores. In two dimensions, //, 0.59 for conductive bonds at all shared edges... 

These pioneering simulation studies clearly demonstrate the presence of the percolation threshold for water under ordinary conditions. Although some polygonal closures can exist, the critical percolation threshold is apparently well predicted by Flory s theory of the gel point. 

Lorenz CD, Ziff RM (2001) Precise determination of the critical percolation threshold for the three-dimensional Swiss cheese model using a growth algorithm. J Chem Phys 114(8) 3659-3661... 

On the critical percolation threshold p = pc, the cluster distribution becomes... 

Thus, fracture occurs by first straining the chains to a critical draw ratio X and storing mechanical energy G (X — 1). The chains relax by Rouse retraction and disentangle if the energy released is sufficient to relax them to the critically connected state corresponding to the percolation threshold. Since Xc (M/Mc) /, we expect the molecular weight dependence of fracture to behave approximately as... 

Above a critical hller concentration, the percolation threshold, the properties of the reinforced rubber material change drastically, because a hller-hUer network is estabhshed. This results, for example, in an overproportional increase of electrical conductivity of a carbon black-hUed compound. The continuous disruption and restorahon of this hller network upon deformation is well visible in the so-called Payne effect [20,21], as represented in Figure 29.5. It illustrates the strain-dependence of the modulus and the strain-independent contributions to the complex shear or tensUe moduli for carbon black-hlled compounds and sUica-hUed compounds. 

The configuration of a system in which percolation may occur is classically treated as one in which the ingredients do not move. Considerable work has been devoted to these static models, leading to numerical solutions of the critical concentrations and cluster sizes associated with a percolation threshold. 

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

The main conclusion of the percolation theory is that there exists a critical concentration of the conductive fraction (percolation threshold, c0), below which the ion (charge) transport is very difficult because of a lack of pathways between conductive islands. Above and near the threshold, the conductivity can be expressed as ... 

With the U-Type systems (i.e. with the low chain alcohols) the trends in the conductivity - curve are consistent with percolative conduction originally proposed to explain the behaviour of conductance of conductor-insulator composite materials (27). In the latter model, the effective conductivity is practically zero as long as the conductive volume fraction is smaller than a critical value called the percolation threshold, beyond which k suddenly takes a non-zero value and rapidly increases with increase of Under these conditions. 

Lastly, Argyrakis and Kopelman [33] have simulated A + B -4 0 and A + A —> 0 reactions on two- and three-dimensional critical percolation clusters which serve as representative random fractal lattices. (The critical thresholds are known to be pc = 0.5931 and 0.3117 for two and three dimensions respectively.). The expected important feature of these reactions is superuniversality of the kinetics independent on the spatial dimension and... 

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